Solutions Near Singular Points to the Eikonal and Related First Order Non-linear Partial Differential Equations in Two Independent Variables

TL;DR

This paper investigates solutions to the eikonal and related first order PDEs near singular points where traditional methods fail, revealing non-uniqueness and irregularity confined to specific curves, using symplectic geometry and Hamiltonian normal forms.

Contribution

It provides a detailed analysis of solution behavior near singular points for the eikonal equation, highlighting non-uniqueness and regularity loss using advanced geometric tools.

Findings

Solutions are non-unique near singular points.

Regularity loss occurs along specific curves through the singularity.

Solutions can be less regular than the initial data and the function H.

Abstract

A detailed study of solutions to the first order partial differential equation H(x,y,z_x,z_y)=0, with special emphasis on the eikonal equation z_x^2+z_y^2=h(x,y), is made near points where the equation becomes singular in the sense that dH=0, in which case the method of characteristics does not apply. The main results are that there is a strong lack of uniqueness of solutions near such points and that solutions can be less regular than both the function H and the initial data of the problem, but that this loss of regularity only occurs along a pair of curves through the singular point. The main tools are symplectic geometry and the Sternberg normal form for Hamiltonian vector fields.